Interdependence of additivity and sine additivity

Abstract

Let S be a semigroup and K a field. A function \(f:S \rightarrow K\) is additive if \(f(xy) = f(x) + f(y)\) for all \(x,y \in S\), and functions \(g,h:S \rightarrow K\) form a sine pair if they satisfy the sine addition law \(g(xy) = g(x)h(y) + h(x)g(y)\) for all \(x,y \in S\). Adding these two equations we arrive at the functional equation (*) \(f(xy) + g(xy) = f(x) + f(y) + g(x)h(y) + h(x)g(y)\). The alienation question for additivity and sine additivity asks whether (*) implies that f is additive and (g, h) is a sine pair. To fully answer this question we find the general solution of (*) for unknown functions \(f,g,h:S \rightarrow {\mathbb {C}}\). The solution illustrates a significant amount of interdependence between additivity and sine additivity.